#6388850

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,26,25)(23,30,27,28)(24,32,34,36)(29,35,33,31);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,26,27)(22,28,25,30)(24,31,34,35)(29,32,33,36);(21,24,26,34)(22,29,25,33)(23,31,27,35)(28,32,30,36);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,25,26,22)(23,30,27,28)(24,33,34,29)(31,36,35,32);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(22,25)(28,30)(29,33)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296046);